Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations

Abstract

Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated Painlevé expansion approach and the Möbious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painlevé waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures.

A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

“…We have some progress in studying of a mixed problem where we come to a necessity of using of the declared matrix BA function. We believe that results of the paper will be useful for further development of the results obtained, for example, in [27,40,42,52,53] and for an investigation of rogue waves (about them see, e.g., [4,5,28,55]) to the Maxwell-Bloch equations.…”

“…We cite here only a small number of pioneering papers relating to the Maxwell-Bloch equations. Some reviews on an application of inverse scattering transform to the MB equations can be found in [1,2,27,40], and for the reduced Maxwell-Bloch equations in [28,63].…”

Darboux transformations and solutions of nonlocal Hirota and Maxwell–Bloch equations

“…Here, we take the symmetric form ( , ) = * (− , − ) = * (− , − ), ( , ) = * (− , − ), as an example to illustrate. Substituting the above symmetric forms into Equation (12), there are…”

“…For the above equations, if the Maxwell equation and the Bloch equation coupling, then we can obtain the Maxwell-Bloch (MB) system. 12 By combining the NLS equation and the MB system together, then we will get the nonlinear Schrödinger and the Maxwell-Bloch (NLS-MB) system. 13,14 Accordingly, on the basis of the NLS equation, we can choose appropriate self-steepening and self-frequency effects, then the NLS-MB system can be reduced to a coupling system of the H-MB system.…”

Modulation instability, conservation laws and localized waves for the generalized coupled Fokas-Lenells equation

“…Breather [9,10] not only has a periodic structure in a particular direction, but also can be used to explain the phenomenon of rogue wave. rogue wave [11][12][13][14][15] is dual localized structures in both time evolution and spatial distribution directions, which comes and disappears without a trace [16]. The mechanism of rogue wave can be regarded as the high amplitude wave, which generated by the collision of soliton and breather [17].…”